Regional problem · Logic and sets

Let \(\varrho\) be a binary relation on the set \(\mathbb{N}\) defined, for all \(x, y \in \mathbb{N}\), by

x \varrho y \iff (\exists k \in A)\ x + 2y = 3k \cdot x.

In each of the cases

\text{(a) } A = \mathbb{N}, \qquad \text{(b) } A = \mathbb{T}, \qquad \text{(c) } A = \mathbb{Q},

where \(\mathbb{T}\) denotes the set of odd natural numbers, decide whether \(\varrho\) is an order relation on \(\mathbb{N}\) and, if it is, whether it is a total order. Decide also, in each case, whether \(\varrho\) is an equivalence relation on \(\mathbb{N}\) and, if it is, determine its equivalence classes.